On a Multigrid Adaptive Refinement Solver for Saturated Non-Newtonian Flow in Porous Media

  • Willy Dörfler
  • Oleg Iliev
  • Dimitar Stoyanov
  • Daniela Vassileva
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2542)


A multigrid adaptive refinement algorithm for non-Newtonian flow in porous media is presented. The saturated flow of non-Newtonian fluid is described by continuity equation and generalized Darcy law. The resulting second order nonlinear elliptic equation is discretized by finite volume method on cell-centered grid. A nonlinear full-multigrid, full-approximation-storage algorithm is implemented. Singe grid solver, based on Picard linearization and Gauss-Seidel relaxation, is used as a smoother. Further, a local refinement multigrid algorithm on a composite grid is developed. A residual based error indicator is used in the adaptive refinement criterion. A special implementation approach is used, which allows us to perform unstructured local refinement in conjunction with the finite volume discretization. Several results from numerical experiments are presented in order to examine the performance of the solver.

Key words

nonlinear multigrid adaptive refinement non-Newtonian flow in porous media 


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  1. [1]
    Greenkorn, R.: Flow Phenomena in Porous Media. Marcel Dekker, Inc., New York and Basel (1983)Google Scholar
  2. [2]
    Ewing, R., Lazarov, R., Vassilevski, P: Local refinement techniques for elliptic problems on cell-centered grids. I. Error analysis. Mathematics of Computations, 56 (1991) 437–461zbMATHCrossRefMathSciNetGoogle Scholar
  3. [3]
    Hackbusch, W.: Multi-Grid Methods and Applications. Springer-Verlag, Berlin Heidelberg New York Tokyo (1985)zbMATHGoogle Scholar
  4. [4]
    Wesseling, P.: An Introduction to Multigrid Methods. New York, Wiley (1991)Google Scholar
  5. [5]
    McCormick, S.F.: Multilevel Adaptive Methods for Partial Differential Equations. SIAM, Philadelphia (1989)zbMATHGoogle Scholar
  6. [6]
    Ewing, R., Lazarov, R., Vassilevski, P: Local refinement techniques for elliptic problems on cell-centered grids, II: Optimal order two-grid iterative methods. Numer. Linear Algebra Appl., 1 (1994) 337–368zbMATHCrossRefMathSciNetGoogle Scholar
  7. [7]
    Iliev, O., Stoyanov, D.: On a multigrid, local refinement solver for incompressible Navier-Stokes equations. Mathematical Modelling, 13 (2001) 95–106zbMATHMathSciNetGoogle Scholar
  8. [8]
    Iliev, O., Stoyanov, D.: Multigrid-adaptive local refinement solver for incompressible flows. Large-Scale Scientific Computing, eds. Margenov S. et al., Lect. Notes Comp. Sci., 2179 (2001) 361–368Google Scholar
  9. [9]
    Dörfler, W.: A convergent adaptive algorithm for Poisson equation. SIAM J. Numer. Anal., 33 (1996) 1106–1124zbMATHCrossRefMathSciNetGoogle Scholar
  10. [10]
    Dörfler, W., Wilderotter, O: An adaptive algorithm finite element method for a linear Elliptic Equation with Variable Coefficients. Z. Angev. Math. Mech., 80 (2000) 481–491zbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  • Willy Dörfler
    • 1
  • Oleg Iliev
    • 2
  • Dimitar Stoyanov
    • 2
  • Daniela Vassileva
    • 3
  1. 1.Institut für Angewandte Mathematik IIUniversität Karlsruhe (TH)KarlsruheGermany
  2. 2.Fraunhofer Institut für Techno- und Wirtschaftsmathematik (ITWM)KaiserslauternGermany
  3. 3.Institute of Mathematics and InformaticsBulgarian Academy of SciencesSofiaBulgaria

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